2. Waves and the Wave Equation

What is a wave?Forward vs. backward propagating wavesThe one-dimensional wave equation

Phase velocityReminders about complex numbersThe complex amplitude of a wave

What is a wave?In the mathematical sense, a wave is any function that moves.

To displace any function f(x)to the right, just change itsargument from x to x-x0,where x0 is a positive number.

If we let x0 = v t, where v is positive and t is time, then the displacement increases with increasing time.

So f(x-vt) represents a rightward, or forward, propagating wave.

Similarly, f(x+vt) represents a leftward, or backward, propagating wave.

v is the velocity of the wave.

-4 -2 0 2 4 6

f(x)

f(x-1) f(x-2)

f(x-3)

The wave equation in one dimension Later, we will derive the wave equation from Maxwell’s equations. Here it is, in its one-dimensional form for scalar (i.e., non-vector) functions, f.

This equation determines the properties of most wave phenomena, not only light waves.

In many real-world situations, the velocity of a wave depends on its amplitude, so v = v(f). In this case, the solutions can be hard to determine.

Fortunately, this is not the case for electromagnetic waves.

2 2

2 2 2

1 0v

f fx t

water wave air wave earth wave

The wave equation is linear: The principle of “Superposition” holds.

This has important consequences for light waves. It means that light beams can pass through each other without altering each other.

It also means that waves can constructively or destructively interfere.

If f1(x,t) and f2(x,t) are solutions to the wave equation,then their sum f1(x,t) + f2(x,t) is also a solution.

Proof: and

2 2 2 2 2 21 2 1 2 1 1 2 2

2 2 2 2 2 2 2 2 2

1 1 1 0v v v

f f f f f f f fx t x t x t

2 2 21 2 1 2

2 2 2

f f f fx x x

2 2 2

1 2 1 22 2 2

f f f ft t t

What if superposition wasn’t true?That would mean that two waves would interact with each other when passing through each other. This leads to some truly odd behaviors.

strange wave collisions

waves anti-crossing

waves spiraling around each other

The solution to the one-dimensional wave equation

The wave equation has the simple solution:

If this is a “solution” to the equation, it seems pretty vague…Is it at all useful?

First, let’s prove that it is a solution.

where f (u) can be any twice-differentiable function.

,f x t f x vt

Proof that f (x ± vt) solves the wave equation

Write f (x ± vt) as f (u), where u = x ± vt. So and 1ux

vut

f fx u

vf ft u

2 2

22 2vf f

t u

2 2

2 2

f fx u

So and

QED

2 2 2 22

2 2 2 2 2 2

1 1 v 0v v

f f f fx t u u

Substituting into the wave equation:

f f ux u x

f f ut u t

Now, use the chain rule:

The 1D wave equation for light waves

2 2

2 2 0E Ex t

where:

E(x,t) is the electric field is the magnetic permeability is the dielectric permittivity

This is a linear, second-order, homogeneous differential equation.

A useful thing to know about such equations:The most general solution has two unknown constants, which

cannot be determined without some additional information about the problem (e.g., initial conditions or boundary conditions).

And:We might expect that oscillatory solutions (sines and cosines) will

be very relevant for light waves.

1D wave equation: some solutionsWe showed that any twice-differentiable function can be a solution, as long as z and t appear in the right combination.

, 5 3E z t z t So this is a solution:

z

E field amplitude

E(z) at t = 0E(z) at a later time

But these are not really very useful solutions.

6, z tE z t e And this is a solution:

z

E field amplitude

E(z) at t = 0

E(z) at a later time

( , ) cos[ ( c )] sin[ ( c )]E z t B k z t C k z t

( c)kz k t

( , ) cos( ) sin( )E z t B kz t C kz t

1ck

where:

Note: this is the Greek letter omega, ‘’. Do not get it confused with ‘w’.

It is more useful to use cosine- and sine-wave solutions:

A more useful form for the solution

( , ) cos( ) sin( )E z t B kz t C kz t For simplicity, we’ll just use the forward-propagating wave for now, so no .

Now we can rewrite this in another form, using a trigonometric identity:

cos(x–y) = cos(x) cos(y) + sin(x) sin(y)

With this identity, our solution becomes:

E(z,t) = A cos[(kz – t) – ]

Even more useful form for the solution

( , ) cos cos( ) sin sin( )E z t A kz t A kz t

Thus our solution to the wave equation becomes:

A cos() = B and A sin() = C

Instead of the unknown constants B and C, let’s use two different unknown constants, A and . We define them so that:

Definitions: Amplitude and Absolute phase

Absolute phase = 0

position, zat t = 0.

z = 0

Absolute phase = 2/3

A

This is a common way of writing the solution to the wave equation:

, cosE z t A kz t

A = Amplitude (we will see that this is related to the wave’s energy) = Absolute phase (or “initial” phase: the phase when z = t = 0)

Clarification: What does this graph mean?Just as an illustration, here is the plot again for Absolute phase = 0

position, zat t = 0.

z = 0

What are we plotting here? Be sure you understand this.

position, zat t = 0.

z = 0

Amplitude of E field vector

This picture contains no information about which way the E field is pointing! Only about the length of the vector at any point on the z axis, not about its direction.

E

z axis

A function of both z and t

, cosE z t A kz t

Note: if you take a snapshot at any instant of time, the magnitude of E oscillates as a function of z.

And: if you sit at any location z, the magnitude of E oscillates as a function of time.

Definitions

Spatial quantities:

Temporal quantities:

x

Wavelength

wave vector: k = 2/wave number: = 1/ = k

Temporal quantities:

t

Period

angular frequency: = 2/frequency: f = 1/ =

For a given time, t0:

For a given position, x0:

The Velocity

How fast is the wave traveling?

Velocity is a reference distancedivided by a reference time.

In terms of the k-vector, k = 2/ , and the angular frequency, = 2f, this is: c = / k

The velocity is the wavelength / period:

c = / = f

z

The wave moves one wavelength, , in one period, .

Do you need to memorize the value of the speed of light in empty space?

YES.

Approximately: c0 = 3×108 meters / second

Note: this is the only constant I expect you to memorize.

The Phase of a Wave

The phase, , is everything inside the cosine.

E(x,t) = A cos(), where = kx – t –

Don’t confuse “the phase” with “the absolute phase” (or “initial phase”).

The angular frequency and wave vector can be expressed as derivatives of the phase:

= – /t

k = /x

Complex numbers

Consider a point,P = (x,y), on a 2D Cartesian grid.

Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number.

where: 1 j

So instead of using an ordered pair, we write:

P = x + j y

The controversy: i vs j

In this class:We strive to be engineers and use j…

(don’t worry… we won’t see current too often.)

In physics: 1 iand j = current density

In engineering: 1 jand i = current

Euler's Formula

cos sin je j

so the point, P = A cos() + j A sin(), can be written:

P = A exp( j )

Truly one of the most important equations in all the world…

where: A = Amplitude

= Phase

Reminder: |exp( j )| = 1, for any (real) value of .

Waves using complex numbers

We often write these

expressions without the

½, Re, or +c.c.

We have seen that the electric field of a light wave can be written:

E(x,t) = A cos(kx – t – )

Since exp(j) = cos() + j sin(), E(x,t) can also be written:

or

where "+ c.c." means "plus the complex conjugate of everything before the plus sign."

1, exp . .2

E x t A j kx t c c

, Re expE x t A j kx t

Waves using complex amplitudes

, exp

exp, p( ) ex

E x t A j kx t

E x t j kx tA j

How do you know if E0 is real or complex?

Sometimes people use the "~", but not always.So always assume it's complex unless told otherwise.

0, exp E x t E j kx t

0 exp( ) (note the " ~ " symbol) E A j

The resulting "complex amplitude" is:

We can let the amplitude be complex:

where we've separated the constant stuff from the rapidly changing stuff.

0E does not depend on x or t.

The complex amplitude of a waveRemember, nothing measurable ever contains j.

Complex numbers are merely a useful bookkeeping tool for tracking the phase of a quantity. They don’t appear in measurements.

0, exp E x t E j kx t 0 exp E A jwhere

The amplitude of an electric field like this one is a quantity that (in principle) we can measure. It has units: volts/meter.

This measurable quantity is never ever complex.

In principle, we can also measure the phase of this quantity (inradians). This value is also always a real number.

0 E A 0 E